Parodies are "derivative works." You must get permission from the rights holders to release a derivative work. (Laws vary between countries on how you do this. Remember the tempest in a teapot over Weird Al Yankovich's parody "Living in an Amish Paradise"?)

Permission must be obtained to publish a derivative work - performance rights are a bit different, and are normally handled by the venue you are performing in.

Cheers -

Pete